Increasing stair depth?

I bought a new home that had some odd design choices as well as odd remodeling choices.

One of the odd design choices is my stairs to the basement only have a depth of 8.5 inches carpeted.

The basement is finished and is divided into two sections. A "game room area" which is carpeted and a "traffic area" which is laminate. It is my path from the garage and outside door to the upstairs. The laminate is the old style pergo boards and is the classic honey oak.

The carpet will need replacing. One thought would be to just do the whole basement floor. Recarpet and replace the pergo with tile. The problem with that is the basement bath is already tiled so to make it look right I would need to do that too.

The kitchen is at the top of the stairs. And is covered with the newer pergo in what looks like hickory which is really dumb because I have oak floors through the rest of the house. I hate when people put in faux wood floors when tile would look so much nicer in tile.

So the question is what to do with the stairs? I have a bunch of the pergo, same style as in the kitchen. I could do it in that but it would not match the oak in the basement and if I redo the kitchen the stairs would not match the kitchen.

So I am thinking of maybe carpeting the stairs when I redo the "game room area". The question there is how to increase the tread depth? Can I istall a stair nose under carpet?